Areas between curves
•Download as PPTX, PDF•
1 like•349 views
The document describes finding the area between two curves, y=x^2+1 and y=-x^2+2x+1. It involves graphing the equations to find the intersection points that bound the region, and then using the property that area is the difference of the areas under the curves to calculate it as the integral from 0 to 1 of -2x^2+2x dx.
Report
Share
Report
Share
Recommended
Calculus for Kids - Lesson 5
This document discusses calculus concepts like derivatives, position, speed, and acceleration. It provides examples of taking the derivative of position functions to get speed, and taking the derivative of speed functions to get acceleration. The document also profiles Leonhard Euler, an influential 18th century mathematician who made pioneering contributions to calculus and other fields. Euler discovered the remarkable formula eiπ + 1 = 0, which relates five fundamental mathematical constants.
Aptitude
This document contains a series of logic and reasoning puzzles with the answers provided. Some examples include:
1) The name of the fifth son would be Fifty.
2) After taking away 2 apples from the original 3 apples, you would have 1 apple remaining.
3) Dividing 30 by 1/2 and then adding 10 would give the answer of 60.
4) The number that does not belong in the series 1,1,2,3,4,5,8,13,21 is 5 because it is not the result of adding the previous two numbers.
Ch 3 rev trashketball exp logs
This document contains 35 multi-step math word problems involving exponents and logarithms. The problems cover topics such as solving exponential and logarithmic equations, evaluating logarithmic expressions, expanding logarithmic expressions, and modeling exponential growth. Sample solutions to some of the problems are provided.
Graphing quadratic equations
The document discusses finding the vertex of quadratic equations by using the formula x = -b/2a and then substituting that x-value back into the original equation to find the corresponding y-value. It provides an example of finding the vertex (2,3) of the equation y=-x^2 + 4x - 1. It then asks the reader to find and graph the vertices of three additional quadratic equations.
Index Notation
The document discusses index notation and how it is used to represent repeated multiplication. It covers the basic rules for multiplying and dividing terms with the same base, including adding/subtracting the indices. It also discusses zero and negative indices, and how numbers raised to the power of 0 or negative powers can be evaluated. Key rules covered are a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), a^0 = 1, and a^-n = 1/a^n.
Factoring by Gemma Maniago
This document provides materials on factoring polynomials for 8th grade mathematics. It includes an overview explaining that factoring is the reverse of multiplication. It then presents different types of factorizable polynomials like common monomial factors and quadratic trinomials. Several activity and assessment cards are included to guide students in matching factors to products, finding perfect square trinomials, and factoring polynomials. It aims to help students understand and apply the concept of factoring through hands-on practice. Answer keys are also provided for student self-checking. In the end, a scoring guide and reflection questions are given to evaluate learning.
Weekly Dose 3 - Maths Olympiad Practice
There are 5 trucks carrying various tons of loads. Truck E carries 1 ton more than the average of all trucks. The document shows the calculation to find that truck E carries 5 tons.
Applications of ode and matrices soluti0n
This document contains examples and explanations of exponential growth, half-life, falling objects, encryption, decryption, and calculating the area of a triangle. It provides sample problems, solutions, and step-by-step workings for each topic. Key concepts include how populations grow exponentially over time, how radioactive materials decay over half-lives, how velocity changes with falling objects, how to encrypt and decrypt messages using matrices, and how to use determinants to calculate the area of a triangle from its vertex coordinates.
Recommended
Calculus for Kids - Lesson 5
This document discusses calculus concepts like derivatives, position, speed, and acceleration. It provides examples of taking the derivative of position functions to get speed, and taking the derivative of speed functions to get acceleration. The document also profiles Leonhard Euler, an influential 18th century mathematician who made pioneering contributions to calculus and other fields. Euler discovered the remarkable formula eiπ + 1 = 0, which relates five fundamental mathematical constants.
Aptitude
This document contains a series of logic and reasoning puzzles with the answers provided. Some examples include:
1) The name of the fifth son would be Fifty.
2) After taking away 2 apples from the original 3 apples, you would have 1 apple remaining.
3) Dividing 30 by 1/2 and then adding 10 would give the answer of 60.
4) The number that does not belong in the series 1,1,2,3,4,5,8,13,21 is 5 because it is not the result of adding the previous two numbers.
Ch 3 rev trashketball exp logs
This document contains 35 multi-step math word problems involving exponents and logarithms. The problems cover topics such as solving exponential and logarithmic equations, evaluating logarithmic expressions, expanding logarithmic expressions, and modeling exponential growth. Sample solutions to some of the problems are provided.
Graphing quadratic equations
The document discusses finding the vertex of quadratic equations by using the formula x = -b/2a and then substituting that x-value back into the original equation to find the corresponding y-value. It provides an example of finding the vertex (2,3) of the equation y=-x^2 + 4x - 1. It then asks the reader to find and graph the vertices of three additional quadratic equations.
Index Notation
The document discusses index notation and how it is used to represent repeated multiplication. It covers the basic rules for multiplying and dividing terms with the same base, including adding/subtracting the indices. It also discusses zero and negative indices, and how numbers raised to the power of 0 or negative powers can be evaluated. Key rules covered are a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), a^0 = 1, and a^-n = 1/a^n.
Factoring by Gemma Maniago
This document provides materials on factoring polynomials for 8th grade mathematics. It includes an overview explaining that factoring is the reverse of multiplication. It then presents different types of factorizable polynomials like common monomial factors and quadratic trinomials. Several activity and assessment cards are included to guide students in matching factors to products, finding perfect square trinomials, and factoring polynomials. It aims to help students understand and apply the concept of factoring through hands-on practice. Answer keys are also provided for student self-checking. In the end, a scoring guide and reflection questions are given to evaluate learning.
Weekly Dose 3 - Maths Olympiad Practice
There are 5 trucks carrying various tons of loads. Truck E carries 1 ton more than the average of all trucks. The document shows the calculation to find that truck E carries 5 tons.
Applications of ode and matrices soluti0n
This document contains examples and explanations of exponential growth, half-life, falling objects, encryption, decryption, and calculating the area of a triangle. It provides sample problems, solutions, and step-by-step workings for each topic. Key concepts include how populations grow exponentially over time, how radioactive materials decay over half-lives, how velocity changes with falling objects, how to encrypt and decrypt messages using matrices, and how to use determinants to calculate the area of a triangle from its vertex coordinates.
Why is (a+b)2 = a2+b2+2ab
This document proves the formula (a+b)2 = a2 + b2 + 2ab using both geometric and algebraic approaches. Geometrically, it represents a+b as a line segment and draws the squares and rectangles that make up (a+b)2. Algebraically, it expands (a+b)2 using the distributive property and collects like terms. Both methods demonstrate that (a+b)2 equals the sum of the squares of a and b plus twice their product.
4_Properties_of_Real_Numbers_HW
This document contains a practice worksheet with 45 problems involving properties of real numbers. The problems include simplifying expressions, replacing symbols to make sentences true, graphing numbers on a number line, finding opposites and reciprocals, and identifying the sets of numbers that describe different variables.
Luas daerah bidang datar (kalkulus integral)
1. The document provides solutions to 4 integral calculus problems calculating the area bounded by curves.
2. The first area calculated is 4.5 units. The second is 2.083 units. The third is approximately 7.3 units. The fourth is approximately 32 units.
3. The document finds the bounding points of each region and calculates the area as a definite integral between those bounds.
2.7 distributive property day 3
The document discusses using the distributive property to find the total area of two adjacent rooms. It provides two examples:
1) Finding the individual areas of each room and adding them together
2) Finding the total length by adding the lengths of each room and multiplying by the common width
The total area found using both methods is 8946 square feet.
Pythagorean theorem and distance formula power point
The document discusses the Pythagorean theorem and how to use it to calculate the length of the hypotenuse of a right triangle. It provides examples of applying the theorem, such as calculating that in a triangle with sides of lengths 3 and 5, the hypotenuse has a length of 5.8. The document also demonstrates using the distance formula to calculate the distance between two points.
O Level A Maths Sec 3 Syallbus 2011 Simultaneous Equation, Indices, Surds, Ex...
O Level A Maths Sec 3 Syallbus 2011
Simultaneous Equation, Indices, Surds, Exponential Equation, Logarithms Worksheet.
Prepared by Musicisintheworld.
Copyright © 2011. All Rights Reserved. No part of this publication may be reproduced without permission from the author, Musicisintheworld.
Algebra paso2
1) The document presents solutions to exercises involving algebraic expressions and polynomials.
2) The exercises include factorizing expressions, performing operations on polynomials, dividing polynomials, solving rational expressions, and determining function domains.
3) The solutions show the step-by-step work and use GeoGebra to verify graphical solutions.
7th math c2 -l41--jan22
1. The document outlines an upcoming math assignment that is due tomorrow, along with two upcoming tests on Friday and an unmentioned date.
2. It provides 5 warm-up math problems covering properties of addition, writing decimals, fractions, solving equations, and simplifying expressions.
3. It introduces the distributive property and provides an example of distributing a number over a sum.
4. It lists 5 additional examples to work through applying concepts like area of a rectangle, evaluating expressions, simplifying expressions using the distributive property, and writing formulas for perimeter of a rectangle.
Factoring Perfect Square Trinomial
1. The document is a lesson plan for teaching factoring perfect square trinomials in Math 8. It includes intended learning outcomes, learning content on factoring perfect square trinomials, sample learning activities involving tiles, and examples to factor.
2. Students will factor perfect square trinomials and find square roots. Learning activities involve using tiles to form squares and noticing patterns. Examples are provided to demonstrate factoring trinomials by getting square roots and forming sums or differences.
3. An evaluation section includes more examples for students to factor completely to check their understanding of factoring perfect square trinomials.
Smart solution
This document provides examples of solutions to problems involving number theory, algebra, geometry, and probability. It contains the following types of problems:
- Number theory word problems involving finding missing terms in sequences
- Solving quadratic equations and determining equations with transformed roots
- Calculating areas and perimeters of geometric shapes like octagons and hexagons
- Finding equations of lines transformed by parallel shifts
- Probability problems involving counting outcomes, finding probabilities of events, and calculating combinations and permutations
The document aims to demonstrate smart solutions to a variety of mathematical problems across different topics through clear explanations and step-by-step working.
Patterns in Sequences
Sequence
Patterns in Sequences
General Term/Rule
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Rmanalogias
The document contains a series of math word problems divided into three levels - Nivel I, Nivel II, and Nivel III. It includes the names of six students and states that the purpose is to practice analogies and numerical distribution. Each problem is presented with the question, potential answers to choose from, and in some cases, the step-by-step work shown. There are over 15 problems presented at each level, focusing on skills like multiplication, division, addition, subtraction, and algebraic expressions.
Weekly Dose 14 - Maths Olympiad Practice
The document contains solutions to 4 math problems:
1) Finding the number of natural numbers less than 1000 where the sum of the first and last digits is 13. The answer is 66.
2) Finding the possible different answers for N using cards numbered 1, 2, 3, 5 placed in boxes. The answer is 5.
3) Finding the ratio of the speed of a bicycle to a car given they travel opposite directions along the same road at steady speeds. The answer is 1:5.
4) Finding the total area of shaded triangles in a pattern of equilateral triangles where each new triangle's unshaded area is 3/4 of the previous triangle. The answer is 81/256.
DISTANCE FORMULA (GRADE 10 MATH)
This document contains a math lesson on deriving and using the distance formula. It begins by showing how to derive the formula for finding the distance between two points on a coordinate plane by replacing the coordinates with variables. Examples are then shown of applying the formula to find distances between points with given coordinates. The document emphasizes that the distance formula can be used to find the distance between any two points on a coordinate plane.
Mathematicalgames
(1) The document describes how to play a mathematical puzzle game called Times Hitori and provides an example.
(2) It also contains a cartoon conversation that explains the key properties of real numbers - Euclid's division lemma and the Fundamental theorem of arithmetic.
(3) The Fundamental theorem of arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, regardless of their order.
Expand brackets and_factorise
This document provides instructions on expanding brackets in algebra. It includes examples of expanding single-term and multi-term brackets such as 2(3n - 4) and 5(3a + 2b) - a(2 + 5b). The goal is to be able to multiply out the terms within brackets and then combine like terms. Students are provided examples and practice problems to expand brackets with one or more terms and potentially simplify the results.
Teach secondary school algebra
1. Al-Khwarizmi wrote the first treatise on algebra, Hisab al-jabr w’al-muqabala, in 820 AD, which provided methods for solving equations. The word "algebra" is derived from "al-jabr" in the title, meaning restoration.
2. Pedro Nunes published the first known European translation of Al-Khwarizmi's work in 1567. Nunes demonstrated geometric representations of algebraic concepts like expanding brackets and completing the square.
3. Al-Sijzi proved geometrically in the 10th century that the binomial expansion of (a + b)3 is a3 + 3ab(a + b) +
Solved exercises double integration
This document contains solutions to exercises involving double integration using Cartesian and polar coordinates. It includes 8 exercises with solutions involving double integrals over various regions in 2D planes. The solutions calculate the double integrals using different orders and techniques of integration, including changing to polar coordinates.
solving a trig problem and sketching a graph example problems
We go through (in great detail) a problem involving solving a trig problem and a problem involving sketching a graph
Maths-MS_Term2 (1).pdf
This document provides solutions to 10 math problems from a marking scheme for Class XII. The problems cover a range of calculus and vector topics. Key steps are shown in the solutions. For example, in problem 1, integrals involving logarithmic and trigonometric functions are solved using substitution techniques. In problem 3, vectors are used to prove an identity involving the sum of two unit vectors and the angle between them. Across the solutions, various mathematical concepts are applied concisely to arrive at the answers.
Alg 2 completing the square
The document provides steps for completing the square to solve quadratic equations:
1) Take half the coefficient of the x-term and square it. Add this to both sides of the equation.
2) Simplify the trinomial on the left side into a perfect square.
3) Take the square root of both sides to isolate the variable term.
Examples are provided to demonstrate solving quadratic equations by completing the square.
Assignment For Matlab Report Subject Calculus 2
This document provides the requirements and assignments for a Calculus 2 Matlab report. It includes topics such as: finding partial derivatives of various functions, studying extrema of functions, evaluating double and triple integrals, and calculating mass and centers of mass of solids. Students are divided into groups and will be randomly assigned a topic involving solving concrete problems numerically using Matlab.
More Related Content
What's hot
Why is (a+b)2 = a2+b2+2ab
This document proves the formula (a+b)2 = a2 + b2 + 2ab using both geometric and algebraic approaches. Geometrically, it represents a+b as a line segment and draws the squares and rectangles that make up (a+b)2. Algebraically, it expands (a+b)2 using the distributive property and collects like terms. Both methods demonstrate that (a+b)2 equals the sum of the squares of a and b plus twice their product.
4_Properties_of_Real_Numbers_HW
This document contains a practice worksheet with 45 problems involving properties of real numbers. The problems include simplifying expressions, replacing symbols to make sentences true, graphing numbers on a number line, finding opposites and reciprocals, and identifying the sets of numbers that describe different variables.
Luas daerah bidang datar (kalkulus integral)
1. The document provides solutions to 4 integral calculus problems calculating the area bounded by curves.
2. The first area calculated is 4.5 units. The second is 2.083 units. The third is approximately 7.3 units. The fourth is approximately 32 units.
3. The document finds the bounding points of each region and calculates the area as a definite integral between those bounds.
2.7 distributive property day 3
The document discusses using the distributive property to find the total area of two adjacent rooms. It provides two examples:
1) Finding the individual areas of each room and adding them together
2) Finding the total length by adding the lengths of each room and multiplying by the common width
The total area found using both methods is 8946 square feet.
Pythagorean theorem and distance formula power point
The document discusses the Pythagorean theorem and how to use it to calculate the length of the hypotenuse of a right triangle. It provides examples of applying the theorem, such as calculating that in a triangle with sides of lengths 3 and 5, the hypotenuse has a length of 5.8. The document also demonstrates using the distance formula to calculate the distance between two points.
O Level A Maths Sec 3 Syallbus 2011 Simultaneous Equation, Indices, Surds, Ex...
O Level A Maths Sec 3 Syallbus 2011
Simultaneous Equation, Indices, Surds, Exponential Equation, Logarithms Worksheet.
Prepared by Musicisintheworld.
Copyright © 2011. All Rights Reserved. No part of this publication may be reproduced without permission from the author, Musicisintheworld.
Algebra paso2
1) The document presents solutions to exercises involving algebraic expressions and polynomials.
2) The exercises include factorizing expressions, performing operations on polynomials, dividing polynomials, solving rational expressions, and determining function domains.
3) The solutions show the step-by-step work and use GeoGebra to verify graphical solutions.
7th math c2 -l41--jan22
1. The document outlines an upcoming math assignment that is due tomorrow, along with two upcoming tests on Friday and an unmentioned date.
2. It provides 5 warm-up math problems covering properties of addition, writing decimals, fractions, solving equations, and simplifying expressions.
3. It introduces the distributive property and provides an example of distributing a number over a sum.
4. It lists 5 additional examples to work through applying concepts like area of a rectangle, evaluating expressions, simplifying expressions using the distributive property, and writing formulas for perimeter of a rectangle.
Factoring Perfect Square Trinomial
1. The document is a lesson plan for teaching factoring perfect square trinomials in Math 8. It includes intended learning outcomes, learning content on factoring perfect square trinomials, sample learning activities involving tiles, and examples to factor.
2. Students will factor perfect square trinomials and find square roots. Learning activities involve using tiles to form squares and noticing patterns. Examples are provided to demonstrate factoring trinomials by getting square roots and forming sums or differences.
3. An evaluation section includes more examples for students to factor completely to check their understanding of factoring perfect square trinomials.
Smart solution
This document provides examples of solutions to problems involving number theory, algebra, geometry, and probability. It contains the following types of problems:
- Number theory word problems involving finding missing terms in sequences
- Solving quadratic equations and determining equations with transformed roots
- Calculating areas and perimeters of geometric shapes like octagons and hexagons
- Finding equations of lines transformed by parallel shifts
- Probability problems involving counting outcomes, finding probabilities of events, and calculating combinations and permutations
The document aims to demonstrate smart solutions to a variety of mathematical problems across different topics through clear explanations and step-by-step working.
Patterns in Sequences
Sequence
Patterns in Sequences
General Term/Rule
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Rmanalogias
The document contains a series of math word problems divided into three levels - Nivel I, Nivel II, and Nivel III. It includes the names of six students and states that the purpose is to practice analogies and numerical distribution. Each problem is presented with the question, potential answers to choose from, and in some cases, the step-by-step work shown. There are over 15 problems presented at each level, focusing on skills like multiplication, division, addition, subtraction, and algebraic expressions.
Weekly Dose 14 - Maths Olympiad Practice
The document contains solutions to 4 math problems:
1) Finding the number of natural numbers less than 1000 where the sum of the first and last digits is 13. The answer is 66.
2) Finding the possible different answers for N using cards numbered 1, 2, 3, 5 placed in boxes. The answer is 5.
3) Finding the ratio of the speed of a bicycle to a car given they travel opposite directions along the same road at steady speeds. The answer is 1:5.
4) Finding the total area of shaded triangles in a pattern of equilateral triangles where each new triangle's unshaded area is 3/4 of the previous triangle. The answer is 81/256.
DISTANCE FORMULA (GRADE 10 MATH)
This document contains a math lesson on deriving and using the distance formula. It begins by showing how to derive the formula for finding the distance between two points on a coordinate plane by replacing the coordinates with variables. Examples are then shown of applying the formula to find distances between points with given coordinates. The document emphasizes that the distance formula can be used to find the distance between any two points on a coordinate plane.
Mathematicalgames
(1) The document describes how to play a mathematical puzzle game called Times Hitori and provides an example.
(2) It also contains a cartoon conversation that explains the key properties of real numbers - Euclid's division lemma and the Fundamental theorem of arithmetic.
(3) The Fundamental theorem of arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, regardless of their order.
Expand brackets and_factorise
This document provides instructions on expanding brackets in algebra. It includes examples of expanding single-term and multi-term brackets such as 2(3n - 4) and 5(3a + 2b) - a(2 + 5b). The goal is to be able to multiply out the terms within brackets and then combine like terms. Students are provided examples and practice problems to expand brackets with one or more terms and potentially simplify the results.
What's hot (16)
Pythagorean theorem and distance formula power point
Pythagorean theorem and distance formula power point
O Level A Maths Sec 3 Syallbus 2011 Simultaneous Equation, Indices, Surds, Ex...
O Level A Maths Sec 3 Syallbus 2011 Simultaneous Equation, Indices, Surds, Ex...
Similar to Areas between curves
Teach secondary school algebra
1. Al-Khwarizmi wrote the first treatise on algebra, Hisab al-jabr w’al-muqabala, in 820 AD, which provided methods for solving equations. The word "algebra" is derived from "al-jabr" in the title, meaning restoration.
2. Pedro Nunes published the first known European translation of Al-Khwarizmi's work in 1567. Nunes demonstrated geometric representations of algebraic concepts like expanding brackets and completing the square.
3. Al-Sijzi proved geometrically in the 10th century that the binomial expansion of (a + b)3 is a3 + 3ab(a + b) +
Solved exercises double integration
This document contains solutions to exercises involving double integration using Cartesian and polar coordinates. It includes 8 exercises with solutions involving double integrals over various regions in 2D planes. The solutions calculate the double integrals using different orders and techniques of integration, including changing to polar coordinates.
solving a trig problem and sketching a graph example problems
We go through (in great detail) a problem involving solving a trig problem and a problem involving sketching a graph
Maths-MS_Term2 (1).pdf
This document provides solutions to 10 math problems from a marking scheme for Class XII. The problems cover a range of calculus and vector topics. Key steps are shown in the solutions. For example, in problem 1, integrals involving logarithmic and trigonometric functions are solved using substitution techniques. In problem 3, vectors are used to prove an identity involving the sum of two unit vectors and the angle between them. Across the solutions, various mathematical concepts are applied concisely to arrive at the answers.
Alg 2 completing the square
The document provides steps for completing the square to solve quadratic equations:
1) Take half the coefficient of the x-term and square it. Add this to both sides of the equation.
2) Simplify the trinomial on the left side into a perfect square.
3) Take the square root of both sides to isolate the variable term.
Examples are provided to demonstrate solving quadratic equations by completing the square.
Assignment For Matlab Report Subject Calculus 2
This document provides the requirements and assignments for a Calculus 2 Matlab report. It includes topics such as: finding partial derivatives of various functions, studying extrema of functions, evaluating double and triple integrals, and calculating mass and centers of mass of solids. Students are divided into groups and will be randomly assigned a topic involving solving concrete problems numerically using Matlab.
Area between curves
The document provides steps to calculate the area between two curves by finding the intersection points using "y=y" and then integrating between those points. It works through multiple examples of finding the area between various curve combinations, including quadratic, cubic, and linear functions. The key steps are finding the intersection points using "y=y", setting up the integral as the top curve minus the bottom curve, and integrating between the bounds found from the intersection points.
6.7 quadratic inequalities
This document discusses how to graph and solve quadratic inequalities. It provides steps for graphing quadratic inequalities by sketching the parabola and shading the appropriate region based on a test point. Examples are given of solving quadratic inequalities graphically by determining the portions of the graph above or below the x-axis and obtaining the solution intervals. Exercises are also worked through to practice solving quadratic inequalities graphically.
MATRICES AND CALCULUS.pptx
This document contains a summary of key topics in multivariable calculus including matrices, differential calculus, functions of several variables, and optimization. Some key points covered include:
- Cayley-Hamilton theorem and its applications
- Finding maxima, minima, and points of inflection for functions of one variable
- Continuity conditions for piecewise functions
- Euler's theorem on homogeneous functions
- Total derivatives and Jacobian matrices
- Taylor series expansions
- Using Lagrange multipliers to optimize functions with constraints
Algebra presentation on topic modulus function and polynomials
Algebra Presentation on Topic Modulus Function and Polynomials
Created by : Jocelyn AS and Alayya MH
Performance Task Binus School Bekasi
Chithra
This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
Engr 213 midterm 2b sol 2009
1) The Wronskian of three functions is evaluated to show they are linearly independent on the interval (1, ∞).
2) The general solution to a nonhomogeneous differential equation is found using the method of undetermined coefficients and applying initial conditions.
3) The method of variation of parameters is used to find a particular solution and the general solution to a nonhomogeneous differential equation.
Algebra 2 Section 3-5
The document discusses solving quadratic equations by completing the square. It defines completing the square as turning a quadratic into a perfect square trinomial that can be factored into a binomial squared. The steps for completing the square are provided. Examples of solving quadratic equations by both using the square root property and completing the square are shown and worked through step-by-step.
C2 st lecture 2 handout
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
48 circle part 1 of 2
This document provides a study package on circles for a mathematics class. It begins with an index listing the topics covered, which include theory, revision, exercises, assertion and reason questions, and past examination questions. It then covers circle theory, equations of circles in various forms including parametric and Cartesian, intercepts made by circles on axes, the position of points with respect to circles, lines and circles, and tangents to circles. Examples are provided to illustrate each concept. The document is intended to be a comprehensive resource for students to learn about circles.
0 calc7-1
The definite integral is used to find the area of a region bounded by curves. Specifically, the area A between two curves f(x) and g(x) on the interval [a,b] is given by the definite integral:
A = ∫ab [f(x) - g(x)] dx
To calculate the area, one finds the points where the two curves intersect, then evaluates the integral of the top curve minus the bottom curve between the bounds a and b. For examples where the region is bounded along the y-axis, the integral is evaluated with respect to y. The definite integral provides a way to precisely calculate the area of a region defined by curves.
8-6 Solving Rational Functions
This document discusses solving rational equations and functions. It provides examples of solving rational equations by multiplying both sides by the LCD, checking for extraneous solutions, and solving word problems involving rates and distances using rational functions. It also gives an example of solving a word problem about a round trip flight with headwinds and tailwinds to determine the wind speed.
Algebra lesson 4.2 zeroes of quadratic functions
This document provides information about quadratic functions and solving for their zeroes (x-intercepts). It discusses factoring quadratic expressions, using the zero product property to set each factor equal to zero. It also introduces the quadratic formula as a way to solve quadratic equations that are not factorable. There is an example of using the quadratic formula to find the zeroes of the function f(x)=x^2 - 3x - 1. The document concludes with practice problems for students to solve for the zeroes of various quadratic functions.
The Mr. K question
The document describes a plan to distract a teacher, Mr. K, in order to steal his coffee. It involves throwing his block of wood in the hallway so he would discover a smart board with tricky math questions. While Mr. K was focused on fixing errors in the answers, the students were able to steal his coffee. The smart board then provides the correct solutions to the math questions to further distract Mr. K.
Assign1 21314-sol
This document contains solutions to 5 calculus assignment problems:
1) Finding values of x such that an inequality is satisfied, with the solution being 3 < x < 4.
2) Proving that the square of any odd integer is odd.
3) Proving there is no rational number whose square is 2.
4) Determining the value of x where a quadratic equation has a minimum, which is x = 1.
5) Proving an equation involving sums and powers using mathematical induction, for all natural numbers n and where r ≠ 1.
Similar to Areas between curves (20)
solving a trig problem and sketching a graph example problems
solving a trig problem and sketching a graph example problems
Algebra presentation on topic modulus function and polynomials
Algebra presentation on topic modulus function and polynomials
More from pmakunja
How to write a good studies project
To write a good studies project, one must have a good plan, collect data well and present it clearly. They should avoid poor planning, data collection, and presentation. Key aspects include having a solid plan, gathering quality data, and showing results effectively.
Intro mathematical studies ia
The mathematical studies project enables students to demonstrate their mathematical skills and knowledge through personal research and investigation of a topic of their choosing. The project should be self-explanatory and understandable to non-mathematicians. Students are assessed on introducing their topic, collecting and analyzing data, using appropriate mathematical processes, interpreting results, structurally communicating their findings, and using proper notation. Suggested project ideas relate mathematical concepts to other fields like biology, economics, and physics. The goal is for students to pursue topics they find interesting while developing research, problem-solving, and communication abilities.
Test help for SAT mathematics
The document provides strategies for maximizing scores on the SAT math section. It discusses the structure of the math test, including three sections of varying difficulty. It emphasizes pacing oneself to miss around 5-6 questions per section. Calculators can only calculate and thinking is required. Common mistakes include careless errors. Guessing can be beneficial if an answer can be eliminated. The strategy of "plugging in" numbers for variables is one of the most powerful techniques, turning algebra problems into arithmetic. Examples are provided to illustrate plugging in.
Introduction to the statistics project
This document provides guidelines for a mathematics statistics project. The project requires students to organize and present information using tables, graphs, diagrams and appropriate notation. Students must demonstrate understanding of practical mathematics applications and use technology. They should use appropriate statistical methods, form a logical argument supported by evidence, and analyze personal research within a 1500 word limit. The project introduction should describe the topic and steps. Students must collect and logically organize relevant data, perform both simple and complex mathematical processes, interpret results, discuss validity, and communicate their work clearly. Example project ideas are provided.
How to write a good stats project
To write a good statistics project, one must have a good plan for the project, collect data well and present it clearly. They should avoid poor planning, data collection, and presentation. The project should use proper mathematical processes.
More from pmakunja (6)
Recently uploaded
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcclinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxMohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Advanced Java[Extra Concepts, Not Difficult].docx
This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.Top five deadliest dog breeds in America
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance
How to Build a Module in Odoo 17 Using the Scaffold Method
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Digital Artifact 1 - 10VCD Environments Unit
Digital Artifact 1 - 10VCD Environments Unit - NGV Pavilion Concept Design
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS Template 2023-2024 by: Irene S. Rueco
A Strategic Approach: GenAI in Education
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Recently uploaded (20)
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptx
How to Build a Module in Odoo 17 Using the Scaffold Method
How to Build a Module in Odoo 17 Using the Scaffold Method
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Pride Month Slides 2024 David Douglas School District
Pride Month Slides 2024 David Douglas School District
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
Areas between curves
- 1. Exploring areas between curves Task Three
- 2. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 1: Graph the equations.
- 3. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 1: Graph the equations.
- 4. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 2: Find the intersections to determine the x values which bound the region of the unknown area. (1, 2) (0, 1)
- 5. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 3: Use logic to determine the best way to get the area. (1, 2) (0, 1) Area of Yellow = Area of Blue – Area of Purple within 0≤𝑥≤1
- 6. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 4: SOLVE!
- 7. THE SPECIAL PROPERTY Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. 𝒂𝒃𝒇𝒙±𝒈𝒙𝒅𝒙=𝒂𝒃𝒇𝒙𝒅𝒙±𝒂𝒃𝒈𝒙𝒅𝒙 Area = (Area of Blue – Area of Purple) Area =𝑎𝑏𝑓𝑥𝑑𝑥−𝑎𝑏𝑔𝑥𝑑𝑥 Area =𝑎𝑏𝑓𝑥±𝑔𝑥𝑑𝑥 Area=𝑎𝑏((−𝑥2+2𝑥+1) − (𝑥2+1))𝑑𝑥 Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥
- 8. THE SPECIAL PROPERTY Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. *CHECK IF ANSWERS ARE CONSISTENT. Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥